Weighted Median Value
Description
Computes a weighted median of a numeric vector.
Usage
weightedMedian(x, w = NULL, idxs = NULL, na.rm = FALSE,
interpolate = is.null(ties), ties = NULL, ...)
Arguments
x |
vector of type integer,
numeric, or logical.
|
w |
a vector of weights the same length as x giving the weights
to use for each element of x. Negative weights are treated as zero
weights. Default value is equal weight to all values.
|
idxs |
A vector indicating subset of elements to
operate over. If NULL, no subsetting is done.
|
na.rm |
a logical value indicating whether NA
values in x should be stripped before the computation proceeds, or
not. If NA, no check at all for NAs
is done.
|
interpolate |
If TRUE, linear interpolation
is used to get a consistent estimate of the weighted median.
|
ties |
If interpolate == FALSE, a character string specifying
how to solve ties between two x's that are satisfying the weighted
median criteria. Note that at most two values can satisfy the criteria.
When ties is "min" ("lower weighted median"), the smaller
value of the two is returned and when it is "max" ("upper weighted
median"), the larger value is returned. If ties
is "mean", the mean of the two values is returned. Finally, if
ties is "weighted" (or NULL) a weighted
average of the two are returned, where the weights are weights of all values
x[i] <= x[k] and x[i] >= x[k], respectively.
|
... |
Not used.
|
Value
Returns a numeric scalar.
For the n elements x = c(x[1], x[2], ..., x[n]) with positive
weights w = c(w[1], w[2], ..., w[n]) such that sum(w) = S, the
weighted median is defined as the element x[k] for which the
total weight of all elements x[i] < x[k] is less or equal to
S/2 and for which the total weight of all elements x[i] > x[k]
is less or equal to S/2 (c.f. [1]).
When using linear interpolation, the weighted mean of x[k-1] and
x[k] with weights S[k-1] and S[k] corresponding to the
cumulative weights of those two elements is used as an estimate.
If w is missing then all elements of x are given the same
positive weight. If all weights are zero, NA_real_ is
returned.
If one or more weights are Inf, it is the same as these weights have
the same weight and the others have zero. This makes things easier for cases
where the weights are result of a division with zero.
If there are missing values in w that are part of the calculation
(after subsetting and dropping missing values in x), then the final
result is always NA of the same type as x.
The weighted median solves the following optimization problem:
α∗=argαmin∑i=1nwi∣xi−α∣
where
x=(x1,x2,…,xn) are scalars and
w=(w1,w2,…,wn) are the corresponding "weights" for each
individual x value.
Author(s)
Henrik Bengtsson and Ola Hossjer, Centre for Mathematical Sciences,
Lund University. Thanks to Roger Koenker, Econometrics, University of
Illinois, for the initial ideas.
References
[1] T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to
Algorithms, The MIT Press, Massachusetts Institute of Technology, 1989.
See Also
median, mean() and
weightedMean().
Examples
x <- 1:10
n <- length(x)
m1 <- median(x)
m2 <- weightedMedian(x)
stopifnot(identical(m1, m2))
w <- rep(1, times = n)
m1 <- weightedMedian(x, w)
m2 <- weightedMedian(x, ties = "weighted")
m3 <- weightedMedian(x, ties = "min")
m4 <- weightedMedian(x, ties = "max")
stopifnot(identical(m1, m2))
w[1] <- 5
m1 <- weightedMedian(x, w)
y <- c(rep(0, times = w[1]), x[-1])
m2 <- median(y)
stopifnot(identical(m1, m2))
w[1] <- 8.5
weightedMedian(x, w)
w[1] <- Inf
weightedMedian(x, w)
w[1] <- 1
w[n] <- Inf
weightedMedian(x, w)
w <- rep(0, times = n)
weightedMedian(x, w)
bench <- function(N = 1e5, K = 10) {
x <- rnorm(N)
gc()
t <- c()
t[1] <- system.time(for (k in 1:K) median(x))[3]
t[2] <- system.time(for (k in 1:K) weightedMedian(x))[3]
t <- t / t[1]
names(t) <- c("median", "weightedMedian")
t
}
print(bench(N = 5, K = 100))
print(bench(N = 50, K = 100))
print(bench(N = 200, K = 100))
print(bench(N = 1000, K = 100))
print(bench(N = 10e3, K = 20))
print(bench(N = 100e3, K = 20))